7 Answers
7

You should also be able to say 'Are I not?' instead of the typical 'Aren't I?' Presumably these colloquialisms result from the fact that neither 'Do you not know?' nor 'Am I not?' have a contraction that is at once easily pronounceable and logical.

The raising of the negative from the subordinate verb "know" to the main verb "do" is known by (some) linguists as (variously) not hopping, negative transportation, and neg raising, and there's at four decades' worth of discussion as to what its rules are. (This includes such things as what verbs are neg-raising verbs in various languages.) Of course, when expanding it back out you have to "lower" the negative back down to the subordinate verb again.

Whereas do not consists of two separate words, and only the "do" part is the "verb" for the purposes of inversion, in couldn't, shouldn't, don't, we can say that -n't is essentially a clitic, i.e. not a "word in its own right". (On phonological grounds, we might end up saying that don't is some kind of "fused" form standing in for a verb plus clitic.) A clitic is dependent upon a particular word, so that generally when that word moves, so does the clitic(s) dependent on it (like the "object pronouns" in Romance languages if you're familiar with them).

Technically one could dissolve the contraction into "Do not you know?" but this would be an awkward construction, because English favors a negative after the subject and before the verb in this interrogative form, as in "Do you not know?"

Even this sounds a bit cumbersome given the contracted alternative. It may be used for emphasis, however, as in the following sentence:

Just to add some historical and lexicographic details. (the daily update is just too slow)

A hint is it isn’t doont, but doe-nt rhyming with toe.
don't and other auxiliary-clitic combinations are no longer ‘the sum of their parts’. They have diverged some centuries ago. It’s more obvious with won't, ain't, mustn't (mussnt).

There's no synchronic way to tell why it isn’t willn’t, mustnt or doont - other than learning those forms, consciously or unconsciously.

Good question, it’s one of the points that shows grammar isn’t as simple as a ‘technical’ or ‘logical’ combination of strings of phonemes (or letters if you prefer so).

The line of thought JdeBP hints to is right, but only if you believe in (sorry, I mean it like that) underlying structures, which often require a lot of language-specific fiddling, beating the purpose of descriptions of Language-in-General.

If you consider don't a contraction of do not, then you need to posit an ordering rule:

Rule: Contraction with not takes place before subject-auxiliary inversion.

Following this rule, with contraction:

You do not know.

You don't know. ← not contraction

Don't you know? ← subject-auxiliary inversion

And without contraction:

You do not know.

Do you not know? ← subject-auxiliary inversion

So the forms Don't you know? and Do you not know? are in alternation. Your alternative form *Do not you know? is ungrammatical, a violation of this rule. This is a common error among non-native speakers.

Occasionally we do find sentences beginning with (aux) + not + (subject) in very formal English, typically in writing:

Will not the electorate of this country consider that they have a right to know these facts?

But this can be explained as an application of Heavy NP Shift occurring after inversion, pushing the long and "heavy" subject the electorate of this country past the negator not:

The electorate of this country will not consider…

Will the electorate of this country not consider… ← subject-auxiliary inversion

Will not the electorate of this country consider… ← heavy NP shift

This unacceptable for relatively short and "light" subjects:

You do not know.

Do you not know? ← subject-auxiliary inversion

*Do not you know? ← ungrammatical; you is not "heavy" enough to shift past not

Although it's clear that -n't was etymologically a clitic, a reduced form of not, Zwicky and Pullum showed in 1983 that it should now be considered an inflectional affix. See their paper, Cliticization vs. Inflection: English N'T, which is freely available online. If you consider don't a negative form of do rather than a (synchronic) contraction, then no ordering rule is necessary; this is the approach taken in The Cambridge Grammar of the English Language.

The electorate example above is borrowed from the cited paper, which in turn borrows the example from Zwicky's 1969 paper Phonological constraints in syntactic description.